The ratio of the total surface area to the lateral surface area of a cylinder whose radius is 20 cm and height 60 cm, is

Options

A. `2:1`
B. `3:2`
C. `4:3`
D. `5:3`

Answer

To find the ratio of the total surface area to the lateral surface area of a cylinder with a radius of 20 cm and a height of 60 cm, we can follow these steps:

### Step 1: Identify the formulas
The total surface area (TSA) of a cylinder is given by the formula:
\[ \text{TSA} = 2\pi r(h + r) \]
The lateral surface area (LSA) of a cylinder is given by the formula:
\[ \text{LSA} = 2\pi rh \]

### Step 2: Substitute the values
Given:
- Radius \( r = 20 \) cm
- Height \( h = 60 \) cm

Now, we can substitute these values into the formulas.

### Step 3: Calculate the total surface area (TSA)
\[ \text{TSA} = 2\pi r(h + r) = 2\pi (20)(60 + 20) \]
\[ = 2\pi (20)(80) \]
\[ = 2\pi (1600) \]
\[ = 3200\pi \, \text{cm}^2 \]

### Step 4: Calculate the lateral surface area (LSA)
\[ \text{LSA} = 2\pi rh = 2\pi (20)(60) \]
\[ = 2\pi (1200) \]
\[ = 2400\pi \, \text{cm}^2 \]

### Step 5: Find the ratio of TSA to LSA
Now, we can find the ratio:
\[ \text{Ratio} = \frac{\text{TSA}}{\text{LSA}} = \frac{3200\pi}{2400\pi} \]
The \(\pi\) cancels out:
\[ = \frac{3200}{2400} = \frac{32}{24} = \frac{4}{3} \]

### Final Answer
Thus, the ratio of the total surface area to the lateral surface area of the cylinder is:
\[ \frac{4}{3} \]